1. Field of the Invention
The present invention is directed to remote surface-orientation measurement. It is applicable particularly, but not exclusively, to passive, monocular systems for making such measurements.
2. Background Information
In the discussion that follows, the following bracketed codes will be used to refer to the sources identified next to them:
[BornandWolf1965] Principles of Optics, M. Born and E. Wolf, Pergamon Press, (1965).
[Collett1993] Polarized Light, E. Collett, Marcel Dekker, (1993).
[Hapke1993] Theory of Reflectance and Emittance Spectroscopy, B. Hapke, Cambridge University Press, (1993).
[Iannarilli2000] xe2x80x9cSnapshot LWIR hyperspectral polarimetric imager for ocean surface sensingxe2x80x9d, F. J. Iannarilli, J. A. Shaw, S. H. Jones, and H. E. Scott, Proc. SPIE Vol 4133 (2000).
[IannarilliRubin2002] xe2x80x9cFeature selection for multi-class discrimination via mixed-integer linear programming, F. J. Iannarilli and P. A. Rubin, submitted (2002).
[Joubert1995] xe2x80x9c3-D surface reconstruction using a polarization state analysisxe2x80x9d, E. Joubert, P. Miche, and R. Debrie, J. Optics 26, pp.2-8 (1995) (French journal; no longer published).
[Proll2000] xe2x80x9cApplication of a liquid crystal spatial light modulator for brightness adaptation in microscopic topometryxe2x80x9d, K. P. Proll, J. M. Nivet, Ch. Voland, and H. J. Tiziani, Applied Optics 39(34), 6430-35 (2000).
[Ward1994] The Optical Constants of Bulk Materials and Films, L. Ward, Institute of Physics Publishing, Philadelphia (1994).
[Windecker1995] xe2x80x9cTopometry of technical and biological objects by fringe projectionxe2x80x9d, R. Windecker and H. J. Tiziani, Applied Optics 34, 3644-3650 (1995).
[Wolff1991] xe2x80x9cConstraining object features using a polarization reflectance modelxe2x80x9d, L. B. Wolff and T. E. Boult, IEEE Trans. on Pattern Analysis and Machine Intelligence, 13(7), 635-657, (1991).
[Wooten1972] Optical Properties of Solids, F. Wooten, Academic Press, (1972).
There are numerous applications for non-contact acquisition of an object""s three-dimensional surface geometry or topometry. Consequently, many types of topometric apparatus have become available, each offering relative strengths and realms of applicability. Optical approaches by nature offer the convenience and spatial resolution compatible with typically desired measurement scales. In 3D photography, retrieval of object topometry is attempted by using multiple viewpoints (stereometry), xe2x80x9cshape-from-Xxe2x80x9d (where xe2x80x9cXxe2x80x9d can be shading, texture and other photometric attributes), or perhaps from illuminant time of flight by employing a synchronized light source and camera shutter. In the similar yet distinct xe2x80x9c3D scanningxe2x80x9d domain, object topometry can be retrieved by using two or more displaced cameras to triangulate a scanned laser-illuminated spot. Structured (incoherent) light or interferometric (coherent) retrieval techniques are also common.
These various approaches can be categorized in accordance with whether the measurement is based on the differential range to the subtended surface element, its stereometric disparity, or its orientation.
Differential range measurements by nature use a controlled (active, directed) illumination source. This category includes the structured light, interferometric, and xe2x80x9ctime-of-flightxe2x80x9d techniques [Windecker1995]. Some of the practical challenges involve control of, or accounting for, spatial non-uniformity of illumination or surface reflectance [Proll2000]. An implicit requirement is the arrangement for pre-measurement setup, so the approach typically cannot be used spontaneously.
Stereometric-disparity approaches implicitly require that feature correspondence be ascertained. That is, they need to identify the point in the image taken from viewpoint A that represents the same real-world point as a given point in the image taken from viewpoint B. Active time-sequential spot illumination (e.g., by laser) overcomes the xe2x80x9cmatchpointxe2x80x9d ambiguity suffered by passive imaging, particularly when surfaces lack features. The effectiveness of spot illumination depends on whether enough light is scattered into the direction of the receiving sensor(s) (stationed at fixed viewpoint), so spot illumination works best for diffusely reflecting surfaces and not so well for glossy (quasi-specular) surfaces. Stereometric-disparity methods cannot be used for objects located outside the overlap in the tandem sensors"" fields of view.
The photometric (xe2x80x9cshape-from-Xxe2x80x9d) measurements are by nature modulated by the object""s surface orientation, which in turn modulates the observable reflectance and surface relief. A strong advantage that such methods (potentially) offer is monocularity, that is, the ability to employ a single sensor from a single viewpoint. This minimizes or eliminates the need for pre-measurement set-up. It thus enables these techniques to be used spontaneously over wide fields of regard. But shape-from-shading and shape-from-texture methods often suffer from ambiguities that typically confine them to ascertaining only relative (differential) rather than absolute surface orientation. One way to overcome the ambiguity is to employ variable-incidence active illumination, but this limits flexibility and measurement spontaneity.
Novel sub-categories of the photometric methods employ radiometric attributes beyond scalar intensity. Perhaps the most evident among these is polarimetry. For ideal smooth surfaces, the Fresnel equations relate the measurable polarimetric attributes, e.g., degree of polarization (xe2x80x9cDoPxe2x80x9d) and angle of polarization (xe2x80x9cAoPxe2x80x9d), to surface orientation. This suggests the possibility of passive monocular 3D imaging, where the light sources are ambient illumination and/or thermal self-emission [Wolff1991, Iannarilli2000]. Specifically, the physics suggests a one-to-one mapping from polarimetric (DoP, AoP) to surface-orientation coordinates.
But closer investigation reveals that, without further refinement, such polarimetric imaging may itself be limited to ascertaining relative rather than absolute surface orientation. Now, by making a simplifying assumption, an approach described in [Joubert1995] does make a polarimetric determination of the surface orientation. But it relies on stereometric measurements (i.e., two or more independent measurements from different vantage points) to determine the angle that the line of sight makes with the surface normal. So far, there has been no reasonably robust way to determine three-dimensional surface orientation by passive polarimetry from a single vantage point.
One reason for this is that the degree of polarization for a given surface orientation can vary due to unknown surface roughness. Attributing the proper absolute scale to DoP is also confounded in thermal-infrared applications became the balance between surface self-emission (i.e., temperature) and reflected environmental illumination is unknown. For example, in an application of monochromatic infrared polarimetry to passive retrieval of ocean surface waveslope [Iannarilli2000], the radiometric balance between self-emission from the warm ocean and reflection from an either cold or warm sky varies the scale of DoP by a factor of 10.
To discuss the problem in more detail, we introduce the notation illustrated by FIG. 1, which is a diagram of the various planes and angles defined by a polarimetric sensor 28 and the surface 12 to be measured. The sensor may have imaging capability, in which case it resolves a point on surface 12 at viewing-angle coordinates (xcex2,xcex1) with respect to its reference frame. We define a right-handed Cartesian sensor reference frame 14 such that the Z-axis is coincident with the sensor""s optical axis. The Z-axis""s positive direction is from the sensor""s entrance aperture to its focal plane, i.e. along the direction of light propagation. If the sensor is non-imaging, the viewing-angle coordinates (xcex2,xcex1) are merely (0,0) by definition.
Angle xcex8 is the angle that the line of sight 16 forms with the measured surface 12""s normal 18. Since the invention to be described below depends on a degree of specular reflection, xcex8 will be referred to as the angle of incidence, and we will refer to the plane that the normal 18 forms with the line of sight 16 as the xe2x80x9csurface plane of incidence.xe2x80x9d The line of sight and the Z axis together define another plane, which we will refer to as the xe2x80x9cbending plane of incidence,xe2x80x9d and the angle between the surface and bending planes of incidence will be referred to as the rotation angle xcfx86. In rest of the discussion, we will refer to (xcex8,xcfx86) as giving the surface""s three-dimensional orientation because, once (xcex8,xcfx86) is determined, the three-space vector {circumflex over (n)}0 normal to the measured surface can be obtained from the viewing-angle coordinates (xcex2,xcex1) by, for instance, solving the system of equations (1):
{circumflex over (l)}xc3x4ŝ=Normalize[Cos xcex1 Sin xcex2, Cos xcex2 Sin xcex1, xe2x88x92Cos xcex1 Cos xcex2]
SolveFor: {circumflex over (n)}
{circumflex over (n)}xc2x7{circumflex over (l)}xc3x4ŝ=xe2x88x92Cos xcex8;
Normalize[{circumflex over (n)}xc3x97(xe2x88x92{circumflex over (l)}xc3x4ŝ)]xc2x7Normalize[{circumflex over (l)}xc3x4ŝxc3x97[0, 0, 1]]=Cos xcfx86;
Normalize[{circumflex over (n)}xc3x97(xe2x88x92{circumflex over (l)}xc3x4ŝ)]xc2x7Normalize[(xe2x88x92{circumflex over (l)}xc3x4ŝ)xc3x97({circumflex over (l)}xc3x4ŝxc3x97[0, 0, 1])]=xe2x88x92Sin xcfx86;
{circumflex over (n)}xc2x7{circumflex over (n)}=1;xe2x80x83xe2x80x83(1)
In Equation (1), the Normalize[ ] operation returns a vector of unit Euclidean length. The operators {xc2x7,xc3x97} are respectively the dot (inner) and cross products. Although Equation (1) provides no closed-form solution, it can be solved by using standard numerical methods (e.g., least-squares Newton optimization).
We digress at this point to note that polarimetric radiance measurements of (incoherent, ergodic) electric-field orientation are by nature invariant to 180xc2x0 coaxial rotation. This implies a 180xc2x0 ambiguity in the polarimetrically inferred angle xcfx86: polarimetric approaches to orientation determination cannot by themselves indicate whether the surface normal points xe2x80x9cupxe2x80x9d or xe2x80x9cdown.xe2x80x9d But this is not a severe limitation, since application-specific considerations will often dispel this ambiguity. For example, one may know a priori that upward-oriented surfaces"" brightnesses are greater than those of downward-oriented surfaces.
To understand the more-severe limitations of previous approaches to polarimetric orientation measurement, it helps to model how the surface under measurement modulates the observable Stokes spectral radiance. The Stokes vector, which is one way of expressing received radiation""s polarization state, is defined as:
I=I(0xc2x0, 0)+I(90xc2x0, 0),
Q=I(0xc2x0, 0)xe2x88x92I(90xc2x0, 0),
U=I(45xc2x0, 0)xe2x88x92I(135xc2x0, 0),
V=I(45xc2x0, 90xc2x0)xe2x88x92I(135xc2x0, 90xc2x0),xe2x80x83xe2x80x83(2)
where I(xcex8,∈) is the intensity of the light vibrations in the direction making an angle xcex8 with the X-axis when the y-component is subjected to a retardation ∈ with respect to the x-component. The literature also uses [s1,s2,s3,s4] or [s0s1,s2,s3] to represent the Stokes parameters.
Equation (3) is a reasonably general model [Hapke1993] for the surface modulation of the measured Stokes radiance spectrum [BornandWolf1965; Collett1993].                                           [                                                                                I                    r                                                                                                                    Q                    r                                                                                                                    U                    r                                                                                                                    V                    r                                                                        ]                    λ                =                                            τ              λ                        ⁢                          xe2x80x83                        ⁢                          (                                                                                                                  k                                                  θ                          ,                          λ                                                                    ⁡                                              [                                                                                                                                            a                                                                  θ                                  ,                                  λ                                                                                                                                                                                    b                                                                  θ                                  ,                                  λ                                                                                                                                                    0                                                                                      0                                                                                                                                                                          b                                                                  θ                                  ,                                  λ                                                                                                                                                                                    a                                                                  θ                                  ,                                  λ                                                                                                                                                    0                                                                                      0                                                                                                                                          0                                                                                      0                                                                                                                      c                                                                  θ                                  ,                                  λ                                                                                                                                                                                    d                                                                  θ                                  ,                                  λ                                                                                                                                                                                                        0                                                                                      0                                                                                                                      -                                                                  d                                                                      θ                                    ,                                    λ                                                                                                                                                                                                                      c                                                                  θ                                  ,                                  λ                                                                                                                                                                    ]                                                              ⁡                                          [                                                                                                                                  I                              i                                                                                                                                                                                          Q                              i                                                                                                                                                                                          U                              i                                                                                                                                                                                          V                              i                                                                                                                          ]                                                        λ                                +                                                      M                    λ                                    ⁡                                      [                                                                                                                        h                                                          θ                              ,                              λ                                                                                                                                                                            0                                                                                                                      0                                                                                                                      0                                                                                      ]                                                  +                                                      L                    λ                                    ⁡                                      [                                                                                                                        1                            -                                                                                          k                                                                  θ                                  ,                                  λ                                                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              a                                                                  θ                                  ,                                  λ                                                                                                                      -                                                          h                              λ                                                                                                                                                                                                                                      -                                                              k                                                                  θ                                  ,                                  λ                                                                                                                      ⁢                                                          xe2x80x83                                                        ⁢                                                          b                                                              θ                                ,                                λ                                                                                                                                                                                                          0                                                                                                                      0                                                                                      ]                                                              )                                +                      [                                                                                P                    λ                                                                                                0                                                                              0                                                                              0                                                      ]                                              (        3        )            
This equation shows that four right-hand-side additive terms contribute to the left-hand-side Stokes radiance. The first term is the quasi-specular Fresnel reflection of the illuminating Stokes radiance [Ii, Qi, Ui, Vi]xcex, which is incident upon the surface from the Snell""s-law direction, xe2x88x92xcex8, and reflected to the sensor viewing the surface at the angle of incidence xcex8. This first term includes an equivalent Mueller matrix for Fresnel reflection [BornandWolf1965; Collett1993]. The scale factor kxcex8,xcex accounts for surface roughness and thus a diminution of the polarizance of an otherwise mirror-smooth surface material.
The second term is the quasi-diffuse reflection of the illuminating panoramic (hemispherical) radiance Mxcex, presumed to be depolarized upon reflection and diffusely scattered in proportion to the surface""s diffuse reflectance hxcex8,xcex. In this model, the terms kxcex8,xcex and hxcex8,xcex are generally coupled, but this point is incidental to our purposes.
The third term is the thermal (Planck) emission; the black-body emission radiance Lxcex is vectorially scaled by the indicated Stokes components, which comply with Kirchhoff""s Law. All of the first three terms represent radiation components that propagate from the object to the sensor and are thus scaled by the line-of-sight path transmittance xcfx84xcex. The fourth term is additive line-of-sight path radiance. To reduce notation, we assume non-polarized line-of-sight path radiance and transmittance.
The Fresnel-reflectance Mueller-matrix elements axcex8,xcex and bxcex8,xcex are respectively the Fresnel average (power) reflectance and disparity, which are functions of optical wavelength xcex and angle of incidence xcex8. The term xe2x80x9cdisparityxe2x80x9d denotes the numerical difference in (power) reflectivity for electric-field polarizations perpendicular and parallel to the plane of incidence. The matrix elements cxcex8,xcex and dxcex8,xcex may be less physically familiar, but they nevertheless represent phase-coherent relationships between incident and reflected perpendicular and parallel polarized-radiance components.
Fresnel Reflectance Factors
Equation (4) presents equations for the Fresnel reflectance factors (as utilized in the Mueller matrix of Equation (3)) as a function of incidence angle xcex8 and the surface""s complex refractive index (xcex7,xcexa):                                           n            λ                    =                                    n              λ                        -                          i              ·                              κ                λ                                                    ⁢                  
                ⁢                              ρ            s                    =                                                    Cos                ⁡                                  [                  θ                  ]                                            -                                                                                          (                                                                        n                          surf                                                                          n                          med                                                                    )                                        2                                    -                                                            Sin                      2                                        ⁡                                          [                      θ                      ]                                                                                                                          Cos                ⁡                                  [                  θ                  ]                                            +                                                                                          (                                                                        n                          surf                                                                          n                          med                                                                    )                                        2                                    -                                                            Sin                      2                                        ⁡                                          [                      θ                      ]                                                                                                          ⁢                  
                ⁢                              ρ            p                    =                                                                                                                (                                                                        n                          surf                                                                          n                          med                                                                    )                                        2                                    -                                                            Sin                      2                                        ⁡                                          [                      θ                      ]                                                                                  -                                                                    (                                                                  n                        surf                                                                    n                        med                                                              )                                    2                                ⁢                                  xe2x80x83                                ⁢                                  Cos                  ⁡                                      [                    θ                    ]                                                                                                                                                                  (                                                                        n                          surf                                                                          n                          med                                                                    )                                        2                                    -                                                            Sin                      2                                        ⁡                                          [                      θ                      ]                                                                                  +                                                                    (                                                                  n                        surf                                                                    n                        med                                                              )                                    2                                ⁢                                  xe2x80x83                                ⁢                                  Cos                  ⁡                                      [                    θ                    ]                                                                                      ⁢                  
                ⁢                              a                          θ              ,                              n                λ                                              =                                                    ρ                s                            ·                              ρ                s                *                                      +                                          ρ                p                            ·                              ρ                p                *                                                    ⁢                  
                ⁢                              b                          θ              ,                              n                λ                                              =                                                    ρ                s                            ·                              ρ                s                *                                      -                                          ρ                p                            ·                              ρ                p                *                                                    ⁢                  
                ⁢                              c                          θ              ,                              n                λ                                              =                                                    ρ                s                            ·                              ρ                p                *                                      +                                          ρ                p                            ·                              ρ                s                *                                                    ⁢                  
                ⁢                              d                          θ              ,                              n                λ                                              =                      i            ⁡                          (                                                                    ρ                    s                                    ·                                      ρ                    p                    *                                                  -                                                      ρ                    p                                    ·                                      ρ                    s                    *                                                              )                                                          (        4        )            
Equation (4) employs complex-variable arithmetic, where i={square root over (xe2x88x921)} and the superscript asterisk (*) denotes complex conjugate. It assumes that the sensor observes the surface of refractive index nsurf though an intervening medium of refractive index nmed [BornandWolf1965]. In what follows, we assume that the intervening medium is air, whose refractive index is 1+0i, so nmed drops out of Equation (4).
Defining xcfx86 and xcex3 by Relating the Surface-Plane-of-Incidence and Sensor Reference Frames
It is important to realize that Equation (3) models the emanating Stokes radiance with respect to the surface plane of incidence. In this reference frame, the angle of polarization corresponding to any particular Stokes spectral sample is measured counter clockwise in FIG. 1 with respect to the plane of incidence""s normal. However, the sensor measures the Stokes radiance with respect to its own coordinate frame. We define the apparent angle of polarization (appAoP) measurable by the Stokes analyzer in the sensor frame 24 as measured counter-clockwise in the X-Y plane with respect to the X-axis. Recall that the AoP is (conventionally) defined as Equation (5) indicates and is not to be confused with the so-called angle of the polarization ellipse:                               AoP          =                                    1              2                        ⁢                          xe2x80x83                        ⁢                          ArcTan              ⁡                              [                                                                            Q                      r                                        ⁢                    Cos                    ⁢                                          xe2x80x83                                        ⁢                    δ                                    ,                                      U                    r                                                  ]                                                    ⁢                  
                ⁢                  δ          =                      ArcTan            ⁡                          [                                                V                  r                                                  U                  r                                            ]                                                          (        5        )            
The transformation from the surface plane of incidence to the sensor reference frame is modeled upon the behavior of a negligibly polarizing collimating optic (e.g., a refractive element operated in the regime of near-normal angles of incidence). It is highly likely that a polarimetric sensor will apply a collimated light beam through its Stokes analyzer.
For purposes of analysis, the collimating optic is envisioned as a virtual xe2x80x9cbending surfacexe2x80x9d that redirects all rays from a given field angle direction (xcex2,xcex1) into a ray bundle along the +Z-axis. This is the basis for using the xe2x80x9cbending plane of incidencexe2x80x9d nomenclature to refer to the plane defined by the line of sight and the Z-axis. It also makes apparent the xe2x80x9cpolarization ray tracingxe2x80x9d that relates Stokes vectors across the reference-frame transformations involved. The result is that the AoP-to-appAoP transformation involves two coaxial Givens rotations. The first, xcfx86, is from the surface-plane-of-incidence frame into the bending-plane-of-incidence frame, and the second, xcex3, is from the latter frame into the sensor X-Y plane. In other words, two offset angles xcfx86 and xcex3 linearly relate AoP to appAoP:
appAoPxe2x88x92AoP=xcex3xe2x88x92xcfx86xe2x80x83xe2x80x83(6)
As was mentioned above, rotation angle xcfx86 is measured counterclockwise from the surface plane of incidence""s normal to that of the bending plane of incidence. Rotation angle xcex3 is measured clockwise to the X-axis from the bending plane of incidence""s normal.
The known angles (xcex2,xcex1) predetermine the line-of-sight vector and consequently the normal to the bending plane of incidence and thus the rotation angle xcex3. Equation (7) relates angle xcex3 to known (xcex2,xcex1). Subscripted variables denote the reference-frame components (x or y) of the indicated vector variable.                                           los            ^                    =                      Normalize            ⁡                          [                                                Cos                  ⁢                                      xe2x80x83                                    ⁢                  α                  ⁢                                      xe2x80x83                                    ⁢                  Sin                  ⁢                                      xe2x80x83                                    ⁢                  β                                ,                                  Cos                  ⁢                                      xe2x80x83                                    ⁢                  β                  ⁢                                      xe2x80x83                                    ⁢                  Sin                  ⁢                                      xe2x80x83                                    ⁢                  α                                ,                                                      -                    Cos                                    ⁢                                      xe2x80x83                                    ⁢                  α                  ⁢                                      xe2x80x83                                    ⁢                  Cos                  ⁢                                      xe2x80x83                                    ⁢                  β                                            ]                                      ⁢                  
                ⁢                  γ          =                      ArcTan            ⁡                          [                                                                    los                    y                                                                                                      los                        x                        2                                            +                                              los                        y                        2                                                                                            ,                                                      -                                          los                      x                                                                                                                          los                        x                        2                                            +                                              los                        y                        2                                                                                                        ]                                                          (        7        )            
(This equation uses two arguments for the arctangent function ArcTan[x,y] to indicate that this is a xe2x80x9cquadrant-awarexe2x80x9d version of ArcTan[y/x]: it distinguishes first- and second-quadrant values from third- and fourth-quadrant ones.) The discussion below uses xcex3""s value to compute xcfx86.
Relating Stokes Vectors between the Surface Plane-of-Incidence and Sensor Reference Frames
In the prior section, we established that a simple linear offset xcex3xe2x88x92xcfx86 relates the apparent angle of polarization (appAoP) measurable in the sensor frame to the AoP that would be measured in the surface plane of incidence. The equivalent Mueller-matrix transformation (Givens rotation) of the Stokes radiance from the surface plane of incidence into the sensor frame (denoted by primed quantities) is [Collett1993]:                               [                                                                      I                  xe2x80x2                                                                                                      Q                  xe2x80x2                                                                                                      U                  xe2x80x2                                                                                                      V                  xe2x80x2                                                              ]                =                                            [                                                                    1                                                        0                                                        0                                                        0                                                                                        0                                                                              Cos                      ⁢                                              xe2x80x83                                            ⁢                      2                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  γ                          -                          φ                                                )                                                                                                                        Sin                      ⁢                                              xe2x80x83                                            ⁢                      2                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  γ                          -                          φ                                                )                                                                                                  0                                                                                        0                                                                                                      -                        Sin                                            ⁢                                              xe2x80x83                                            ⁢                      2                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  γ                          -                          φ                                                )                                                                                                                        Cos                      ⁢                                              xe2x80x83                                            ⁢                      2                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  γ                          -                          φ                                                )                                                                                                  0                                                                                        0                                                        0                                                        0                                                        1                                                              ]                        ⁡                          [                                                                    I                                                                                        Q                                                                                        U                                                                                        V                                                              ]                                =                      R            ⁢                          xe2x80x83                        ⁢                                          (                                  γ                  -                  φ                                )                            ⁡                              [                                                                            I                                                                                                  Q                                                                                                  U                                                                                                  V                                                                      ]                                                                        (        8        )            
Combining Equation (3) and Equation (8) yields an expression, Equation (9), for the Stokes radiance spectrum measured by the sensor:                                           [                                                                                I                    r                    xe2x80x2                                                                                                                    Q                    r                    xe2x80x2                                                                                                                    U                    r                    xe2x80x2                                                                                                                    V                    r                    xe2x80x2                                                                        ]                    λ                =                              R            ⁢                          xe2x80x83                        ⁢                                                                                (                                          γ                      -                      φ                                        )                                    ⁡                                      [                                                                                                                                                                                      τ                                λ                                                            ⁢                                                              {                                                                                                                                            k                                                                              θ                                        ,                                        λ                                                                                                              ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          (                                                                                                                                                                    a                                                                                          θ                                              ,                                              λ                                                                                                                                ⁢                                                                                      xe2x80x83                                                                                    ⁢                                                                                      I                                                                                          i                                              ,                                              λ                                                                                                                                                                      +                                                                                                                              b                                                                                          θ                                              ,                                              λ                                                                                                                                ⁢                                                                                      xe2x80x83                                                                                    ⁢                                                                                      Q                                                                                          i                                              ,                                              λ                                                                                                                                                                                                          )                                                                                                        +                                                                                                            h                                                                              θ                                        ,                                        λ                                                                                                              ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          M                                      λ                                                                                                        +                                                                                                            (                                                                              1                                        -                                                                                                                              k                                                                                          θ                                              ,                                              λ                                                                                                                                ⁢                                                                                      xe2x80x83                                                                                    ⁢                                                                                      a                                                                                          θ                                              ,                                              λ                                                                                                                                                                      -                                                                                  h                                                                                      θ                                            ,                                            λ                                                                                                                                                              )                                                                        ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          L                                      λ                                                                                                                                      }                                                                                      +                                                          P                              λ                                                                                                                                                                                                                                                                      τ                                λ                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              k                                                                  θ                                  ,                                  λ                                                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                                                                                            b                                                                              θ                                        ,                                        λ                                                                                                              ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          I                                                                              i                                        ,                                        λ                                                                                                                                              +                                                                                                            a                                                                              θ                                        ,                                        λ                                                                                                              ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          Q                                                                              i                                        ,                                        λ                                                                                                                                                                            )                                                                                      -                                                                                          τ                                λ                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              k                                                                  θ                                  ,                                  λ                                                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              b                                λ                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              L                                λ                                                                                                                                                                                                                                                                    τ                              λ                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                          k                                                              θ                                ,                                λ                                                                                      ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    c                                                                          θ                                      ,                                      λ                                                                                                        ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      U                                                                          i                                      ,                                      λ                                                                                                                                      +                                                                                                      d                                                                          θ                                      ,                                      λ                                                                                                        ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      V                                                                          i                                      ,                                      λ                                                                                                                                                                  )                                                                                                                                                                                                                                      τ                              λ                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                          k                                                              θ                                ,                                λ                                                                                      ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    -                                                                          d                                                                              θ                                        ,                                        λ                                                                                                                                              ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      U                                                                          i                                      ,                                      λ                                                                                                                                      +                                                                                                      c                                                                          θ                                      ,                                      λ                                                                                                        ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      V                                                                          i                                      ,                                      λ                                                                                                                                                                  )                                                                                                                                            ]                                                  ⁢                                  
                                [                                                                                                    I                        r                        xe2x80x2                                                                                                                                                Q                        r                        xe2x80x2                                                                                                                                                U                        r                        xe2x80x2                                                                                                                                                V                        r                        xe2x80x2                                                                                            ]                            λ                                =                      [                                                                                                                              τ                        λ                                            ⁢                                              {                                                                                                            k                                                              θ                                ,                                λ                                                                                      ⁢                                                          xe2x80x83                                                        ⁢                                                          (                                                                                                                                    a                                                                          θ                                      ,                                      λ                                                                                                        ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      (                                                                                                                  I                                                                                  i                                          ,                                          λ                                                                                                                    -                                                                              L                                        λ                                                                                                              )                                                                                                  +                                                                                                      b                                                                          θ                                      ,                                      λ                                                                                                        ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      Q                                                                          i                                      ,                                      λ                                                                                                                                                                  )                                                                                +                                                                                    h                                                              θ                                ,                                λ                                                                                      ⁢                                                          xe2x80x83                                                        ⁢                                                          M                              λ                                                                                +                                                                                    (                                                              1                                -                                                                  h                                                                      θ                                    ,                                    λ                                                                                                                              )                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                          L                              λ                                                                                                      }                                                              +                                          P                      λ                                                                                                                                                              τ                      λ                                        ⁢                                          xe2x80x83                                        ⁢                                          k                                              θ                        ,                        λ                                                              ⁢                                          {                                                                                                    (                                                                                                                            b                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                            I                                                                              i                                        ,                                        λ                                                                                                              -                                                                          L                                      λ                                                                                                        )                                                                                            +                                                                                                a                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  Q                                                                      i                                    ,                                    λ                                                                                                                                                        )                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      Cos                            ⁡                                                          [                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      γ                                    -                                    φ                                                                    )                                                                                            ]                                                                                                      +                                                                              (                                                                                                                            c                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  U                                                                      i                                    ,                                    λ                                                                                                                              +                                                                                                d                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  V                                                                      i                                    ,                                    λ                                                                                                                                                        )                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      Sin                            ⁡                                                          [                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      γ                                    -                                    φ                                                                    )                                                                                            ]                                                                                                                          }                                                                                                                                                              τ                      λ                                        ⁢                                          xe2x80x83                                        ⁢                                          k                                              θ                        ,                        λ                                                              ⁢                                          {                                                                                                    (                                                                                                                            c                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  U                                                                      i                                    ,                                    λ                                                                                                                              +                                                                                                d                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  V                                                                      i                                    ,                                    λ                                                                                                                                                        )                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      Cos                            ⁡                                                          [                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      γ                                    -                                    φ                                                                    )                                                                                            ]                                                                                                      -                                                                              (                                                                                                                            b                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                                                            I                                                                              i                                        ,                                        λ                                                                                                              -                                                                          L                                      λ                                                                                                        )                                                                                            +                                                                                                a                                                                      θ                                    ,                                    λ                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  Q                                                                      i                                    ,                                    λ                                                                                                                                                        )                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      Sin                            ⁡                                                          [                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  (                                                                      γ                                    -                                    φ                                                                    )                                                                                            ]                                                                                                                          }                                                                                                                                                              τ                      λ                                        ⁢                                          xe2x80x83                                        ⁢                                          k                                              θ                        ,                        λ                                                              ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    -                                                          d                                                              θ                                ,                                λ                                                                                                              ⁢                                                      xe2x80x83                                                    ⁢                                                      U                                                          i                              ,                              λ                                                                                                      +                                                                              c                                                          θ                              ,                              λ                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                      V                                                          i                              ,                              λ                                                                                                                          )                                                                                            ]                                              (        9        )            
Factors Confounding Inference of (xcex8,xcfx86) from the Stokes Radiance Spectrum
Now, Equation (10) compares the number of known quantities in Equation (9) with its number of putatively unknown quantities, and the results make attempting such a computation appear ill-fated. Even if the surface""s complex refractive index nxcex and therefore Fresnel reflectance factors axcex8,nxcex, bxcex8,nxcex, cxcex8,n,xcex, dxcex8,nxcex are known, the number of unknown terms still exceeds the knowns.
Knowns: [Ixe2x80x2r Qxe2x80x2r Uxe2x80x2r Vxe2x80x2r]xcex,xcex3
Unknowns:
xe2x86x92Orientation: xcex8,xcfx86;
xe2x86x92Surface Parameters: nxcex,kxcex8,xcex,h74 ,xcex,Lxcex
xe2x86x92Illumination: Mxcex,[Ii,xcex Qi,xcex Ui,xcex Vi,xcex]
xe2x86x92Propagation Path: xcfx84xcex,Pxcexxe2x80x83xe2x80x83(10)
Some workers in this art have simplified the problem by limiting it to conditions in which the surface is illuminated by only unpolarized light (i.e. zero-valued components Qi,xcex, Ui,xcex, and Vi,xcex). Under this assumption, Equation (9) simplifies to Equation (11).                                           [                                                                                I                    r                    xe2x80x2                                                                                                                    Q                    r                    xe2x80x2                                                                                                                    U                    r                    xe2x80x2                                                                                                                    V                    r                    xe2x80x2                                                                        ]                    λ                =                  [                                                                                                                τ                      λ                                        ⁢                                          {                                                                                                    k                                                          θ                              ,                              λ                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                      a                                                          θ                              ,                                                              n                                λ                                                                                                              ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                                                          I                                                                  i                                  ,                                  λ                                                                                            -                                                              L                                λ                                                                                      )                                                                          +                                                                              h                                                          θ                              ,                              λ                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                      M                            λ                                                                          +                                                                              (                                                          1                              -                                                              h                                                                  θ                                  ,                                  λ                                                                                                                      )                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      L                            λ                                                                                              }                                                        +                                      P                    λ                                                                                                                                            τ                    λ                                    ⁢                                      xe2x80x83                                    ⁢                                      k                                          θ                      ,                      λ                                                        ⁢                                      xe2x80x83                                    ⁢                                      b                                          θ                      ,                                              n                        λ                                                                              ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  I                                                  i                          ,                          λ                                                                    -                                              L                        λ                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                                      Cos                    ⁡                                          [                                              2                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      γ                            -                            φ                                                    )                                                                    ]                                                                                                                                                                τ                    λ                                    ⁢                                      xe2x80x83                                    ⁢                                      k                                          θ                      ,                      λ                                                        ⁢                                      xe2x80x83                                    ⁢                                      b                                          θ                      ,                                              n                        λ                                                                              ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  I                                                  i                          ,                          λ                                                                    -                                              L                        λ                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                                      Sin                    ⁡                                          [                                              2                        ⁢                                                  xe2x80x83                                                ⁢                                                  (                                                      γ                            -                            φ                                                    )                                                                    ]                                                                                                                          0                                              ]                                    (        11        )            
From this, one can see that Equation (12) for angle xcfx86 immediately results from forming the ratio of measured Qxe2x80x2r and Uxe2x80x2r. (There is a 90-degree ambiguity in the inferred value of xcfx86, which is discussed later.)                               φ          =                      γ            ±                                          1                2                            ⁢                              xe2x80x83                            ⁢                              ArcCos                ⁡                                  [                                                            ±                                              Q                        r                        xe2x80x2                                                                                                                                      Q                          r                          xe2x80x22                                                +                                                  U                          r                          xe2x80x22                                                                                                      ]                                                                    ⁢                  
                ⁢                  φ          =                                    γ              ±                                                1                  2                                ⁢                                  xe2x80x83                                ⁢                                  ArcTan                  ⁡                                      [                                                                  U                        r                        xe2x80x2                                                                    Q                        r                        xe2x80x2                                                              ]                                                                        +                          {                                                                    0                                                                                        90                                                              }                                                          (        12        )            
The Joubert approach mentioned above exploits this simplification to determine the rotation angle xcfx86, but it resorts to making two or more such xcfx86 measurements from different vantage points to derive xcex8 geometrically.
From examination of Equation (11), one can appreciate how apparently intractable it is to infer angle xcex8 radiometrically by taking measurements from only a single vantage point. Angle-xcex8 retrieval appears confounded because surface roughness and line-of-sight transmittance (parameters kxcex8,xcex and xcfx84xcex.), illumination Ii,xcex, surface thermal emission Lxcex, and refractive index nxcex are unknown.
The present invention provides a way to make a polarimetric orientation determination in a way that is robust to these and other effects. According to the invention, the received radiation""s polarization state is measured at several wavelengths in the neighborhood of a resonance in the target surface""s refractive index. From these measurements, the surface""s orientation is computed by taking advantage of an aspect of the relationship between reflected Stokes radiance and surface orientation.
Now, the fact that the Fresnel equations relate surface orientation to the reflected Stokes radiance is well known to those skilled in the art, as is the distinctive spectral behavior of the complex refractive index across a resonance feature. But I have recognized that this resonance behavior causes the Fresnel reflectance factors to have spectral patterns that change characteristically with angle of incidence. I have recognized further that, as manifested in the measurable Stokes radiance spectrum, these patterns are self-normalizable and thus invariant to nuisance parameters. I have also recognized that the inference problem can be simplifiedxe2x80x94without unduly circumscribing its range of applicabilityxe2x80x94by assuming that elliptical (but not necessarily linear) polarization is negligible in any surface-incident radiation whose reflection contributed to the received radiation.
According to the invention, therefore, the equations for the polarization state are solved for surface orientation by assuming (a) the absence of elliptical polarization in any incident radiation whose reflection contributed to the received radiation and (b) the dependence on angle of incidence that Fresnel reflectance exhibits as a function of wavelength in a refractive-index resonance""s neighborhood. As will be seen below, this yields an orientation inference that is robust to most sources of nuisance variability and whose ambiguities can readily be dispelled by using application-specific information in many cases.